$(<\kappa)$-closure for Prikry forcing

Question

A p.o. $\mathcal{P}$ is $(<\kappa)$-closure, if for every decreasing sequence of $<\kappa$ conditions in the forcing $p_0 \geq p_1 \geq \cdots$, there is a condition that is below all of them. Prikry forcing $\mathcal{P}$ is the set of pairs $(s,A)$ where s is a finite subset of a fixed measurable cardinal $\kappa$, and $A$ is an element of a fixed filter $D$ on $\kappa$. A condition $(s,A)$ is stronger than $(t, B)$ if $t$ is an initial segment of $s$ and $A \cup (s-t) \subseteq B$. Why doesn't Prikry forcing have this property? Could someone help me out with this?

Answer 1

I believe that in Prikry forcing the filter $\mathscr{D}$ is generally taken to be a normal ultrafilter on $\kappa$, though it doesn’t matter here: this partial order clearly isn’t countably closed, since for a fixed $A\in\mathscr{D}$ no strictly descending chain $\langle s_0,A\rangle > \langle s_1,A\rangle > \langle s_2,A\rangle >\ldots$ can be extended: $\bigcup_n s_n$ is infinite.